Following the notation and conventions of [1]}, recall that the choice of \(z\)  gives rise to an infinite cyclic cover \(\pi _z: X_z\rightarrow X\) . Given \(n\in \mathbb {Z}_{\ge 0}\)  denote by \(\pi _z^n: X_{z,p^n}\rightarrow X\)  the unique \(\mathbb {Z}/p^n\mathbb {Z}\) -subcover of \(\pi _z\) . Let \(\xi _{ z}^n:M_{z,p^n}\rightarrow S^3\)  be the Fox completion of \(\pi _{ z}^n\)  (see [2]} for further details). Note that \(\xi _{ z}^n\)  is branched along \(\mathcal {L}\) . Given a \(\mathbb {Z}\) -module \(A\) , set \(|A|\)  to denote the cardinality of \(A\)  if it is finite and 0 if \(A\)  is infinite.
